Relations and Functions - Class 12

Let $S = \{ A \in M_3(\mathbb{R}) \mid A^T A = I \text{ and } \det(A) = 1 \}$. For any matrix $M \in M_3(\mathbb{R})$, its Frobenius norm is defined as $||M|| = \sqrt{\text{Tr}(M^T M)}$. Two relations $\mathcal{R}_1$ and $\mathcal{R}_2$ are defined on $S$ as follows: $A \ \mathcal{R}_1 \ B \iff ||A - B||^2 < p$ $A \ \mathcal{R}_2 \ B \iff ||A - B||^2 > q$ If $p$ is the smallest integer such that $\mathcal{R}_1$ is the universal relation on $S$, and $q$ is the smallest integer such that $\mathcal{R}_2$ is the empty relation on $S$, then the value of $p + q$ is:

Let $X = \{1, 2, 3\}$. A relation $R$ on $X$ is chosen at random from the set of all possible relations on $X$. Let $E_1$ be the event that $R$ is the universal relation on $X$ and is also anti-symmetric. Let $E_2$ be the event that $R$ is an equivalence relation on $X$. Which of the following statements is true regarding the events $E_1$ and $E_2$?

Let $f(x) = \int_0^x (t^2 - 2|t| + 1)^2 (t^2 - 9)^2 dt$. Let $S = \{x \in \mathbb{R} : f'(x) = 0\}$. A relation $R$ is defined on $S$ such that for $a, b \in S$, $a R b$ if and only if $a < b$ and there exists $c \in (a, b)$ satisfying $f''(c) = 0$. Find the number of elements in $R$.

Let $S$ be the set of all points that divide the line segment joining $A(2, -3, 4)$ and $B(-1, 1, -1)$ strictly internally. A relation $R_n$ on $S$ is defined as $(P, Q) \in R_n$ if and only if both $P$ and $Q$ lie strictly inside the $n^{\text{th}}$ octant. Let $E$ be the set of all octant numbers $n \in \{1, 2, \dots, 8\}$ for which $R_n$ is an empty relation. If $P_\alpha$ and $P_\beta$ are the boundary points of the segment of $AB$ that lies in the $3^{\text{rd}}$ octant, find the value of $\sum_{n \in E} n + 9 \times (\text{sum of all coordinates of } P_\alpha \text{ and } P_\beta)$.

Let $S$ be the set of all points on the line $L: \frac{x-2}{1} = \frac{y-1}{2} = \frac{z-3}{2}$. Let $\Pi$ be the plane containing the point $B(1, 0, -1)$ and parallel to the vectors $\vec{p} = \hat{i} + \alpha\hat{j} + \hat{k}$ and $\vec{q} = 2\hat{i} - \hat{j} + \beta\hat{k}$, where $\alpha, \beta \in \mathbb{Z}^+$ and $\alpha < \beta$. A relation $R$ on $S$ is defined as: for $X, Y \in S$, $X R Y$ if and only if the midpoint of the line segment $XY$ lies on the plane $\Pi$. If $R$ is an empty relation on $S$, find the value of $(\alpha + \beta)(\alpha \beta)$.

Let $S$ be the set of all $3 \times 3$ matrices over $\mathbb{R}$ with entries from $\{0, 1\}$. We define three subsets of $S$ as follows: $X = \{ A \in S \mid \det(A) = 0 \text{ and } \det(R_{12}(A)) \neq 0 \}$ $Y = \{ A \in S \mid \det(A) \neq 0 \text{ and } \det(R_{12}(A)) = \det(A) \}$ $W = \{ A \in S \mid A^T = -A \text{ and } A \neq O \}$ where $R_{12}(A)$ denotes the matrix obtained by swapping the first and second rows of $A$, and $O$ is the $3 \times 3$ null matrix. Let $Z = X \cup Y \cup W$. Let $n$ be the number of reflexive relations on $Z$, and $m$ be the number of symmetric relations on $Z$. Find the value of $n + m$.

Let $S = \{ z \in \mathbb{C} \mid z^2 - 2iz - 2 = 0 \text{ and } \bar{z}^2 - 2i\bar{z} - 2 = 0 \}$. Let $R$ be a relation on $S$ defined by $z_1 R z_2 \iff z_1 + z_2 = 0$. Which of the following statements is true?

Let $S$ be the set of all ellipses $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) such that their eccentricity $e$ satisfies $\int_{0}^{e} |3t - 1| \, dt = \frac{5}{6}$ A relation $R$ is defined on $S$ by $E_1 R E_2$ if and only if the area of $E_1$ is strictly less than the area of $E_2$. Which of the following statements is true about the relation $R$?

Let $X = \{1, 2, 3, \dots, 20\}$. For each $x \in X$, define the set $S_x$ as the interval: $S_x = \left[ \left\lfloor \frac{x}{4} \right\rfloor, \left\lfloor \frac{x}{4} \right\rfloor + \log_2\left(1 + \sin^2\left(\frac{\pi x}{2}\right)\right) \right)$ where $\lfloor \cdot \rfloor$ denotes the greatest integer function. A relation $R$ is defined on $X$ such that $x R y$ if and only if $S_x \cap S_y \neq \emptyset$. Which of the following statements is true?

Let $M(x) = \begin{pmatrix} [x] & [x+\frac{1}{3}] & [x+\frac{2}{3}] \\ [x+\frac{1}{3}] & [x+\frac{2}{3}] & [x+1] \\ [x+\frac{2}{3}] & [x+1] & [x+\frac{4}{3}] \end{pmatrix}$, where $[t]$ denotes the greatest integer less than or equal to $t$. Let $S = \{ M(x) \mid x \in [0, 10] \}$. A relation $R$ is defined on $S$ such that $M(x) R M(y) \iff \det(M(x)) = \det(M(y))$. Let $N$ be the number of distinct equivalence classes of $R$, and $I = \int_{0}^{10} |\det(M(x))| \, dx$. Find the value of $N + 3I$.

Define a relation $\sim$ on $\mathbb{R}$ by $\alpha \sim \beta \iff \sin \alpha = \sin \beta$. For $x > 1$, consider the function $S(x) = \arcsin\left(\frac{2x}{1+x^2}\right) + \arccos\left(\frac{1-x^2}{1+x^2}\right) + \arctan\left(\frac{2x}{1-x^2}\right)$ Let $P(x)$ be the unique value in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ such that $P(x) \sim S(x)$. Which of the following expressions is identically equal to $P(x)$ for all $x > 1$?

Let $S$ be the family of equations $E_a : a x^2 + (a^2 + 4) x + 4a = 0$ for $a \in \mathbb{R}$. A relation $R$ is defined on $S$ such that $E_a R E_b$ if and only if $E_a$ and $E_b$ have the exact same set of real roots. Let $[E_a]$ denote the equivalence class of $E_a$ under $R$. Find the number of values of $a \in \mathbb{R}$ for which $[E_a]$ contains exactly one element.

Let $A, B,$ and $C$ be three events in a probability space such that $P(A) = P(B) = P(C) > 0$. Let $I_E$ denote the indicator function of an event $E$. If the relation $|I_A - I_C| = |I_A - I_B| + |I_B - I_C|$ holds almost surely, and $P(A|B) = P(B|C) = \frac{3}{4}$, find the value of $P(A|C)$.

Let $S_5$ be the set of all permutations of the set $\{1, 2, 3, 4, 5\}$. An equivalence relation $\sim$ is defined on $S_5$ such that for $\sigma, \tau \in S_5$, $\sigma \sim \tau$ if and only if $\sigma$ and $\tau$ have the exact same number of fixed points. A permutation $\sigma$ is chosen uniformly at random from $S_5$. Let $[\sigma]$ denote the equivalence class containing $\sigma$. What is the expected value of $|[\sigma]|$, the cardinality of the equivalence class containing $\sigma$?

Let $S$ be the set of all $3 \times 3$ matrices with real entries. A relation $R$ on $S$ is defined by $A R B \iff \det(A) = \det(B)$. Let $[M]$ denote the equivalence class of a matrix $M \in S$. Consider the matrix $P = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix}$. If a matrix $A \in [P]$ satisfies the equation $A \cdot \text{adj}(A) + 3A = A^2 + k I$ for some real number $k$, which of the following statements is ALWAYS true?

Let $S = \{ z \in \mathbb{C} : z^{40} = 1 \}$ and $A = \{ z \in S : |z^2 - 1| \le \sqrt{2} \}$. An equivalence relation $R$ is defined on $A$ by $z_1 R z_2 \iff z_1^4 = z_2^4$. If $N$ is the number of elements in $A$, and $M$ is the number of equivalence classes of $R$ on $A$, find the value of $N + M$.

Let $X = [0, 2\pi]$. A relation $R$ is defined on $X$ by $x R y \iff f(x) = f(y)$, where $f(x) = \cos^{-1}(\cos x) - \left| \sin^{-1}(\sin x) + \frac{\pi}{6} \right|$ Let $E_x$ denote the equivalence class of $x$ under $R$. Let $S$ be the set of all $c \in \mathbb{R}$ such that the equation $f(x) = c$ has exactly $2$ solutions in $X$ (i.e., $|E_x| = 2$ for any $x$ with $f(x) = c$). If $S$ can be expressed as the union of two disjoint open intervals $(a, b) \cup (c, d)$ with $a < b \le c < d$, then the value of $a+b+c+d$ is:

Let $S$ be a multiset containing 12 elements, consisting of 6 identical items of type $X$ and 6 distinct items $Y_1, Y_2, Y_3, Y_4, Y_5, Y_6$. Find the number of ways to partition the multiset $S$ into 6 unordered pairs.

A curve is defined parametrically by $x(t) = t - a \sin t$ and $y(t) = t - b \sin t$, where $t \in \mathbb{R}$ and $a, b \in \mathbb{R}$. The set of all pairs $(a, b)$ for which $y$ can be expressed as a single-valued injective function of $x$ on the entire real line $\mathbb{R}$ is:

Let $S = \{ z \in \mathbb{C} : |z| = 1, z \neq 1 \}$. A function $f: S \to \mathbb{C}$ is defined as $f(z) = z \frac{z-1}{|z-1|}$. Let $N(w)$ denote the number of distinct solutions $z \in S$ to the equation $f(z) = w$. Which of the following statements is TRUE?

The number of real roots of the equation $\tan^{-1}\left(\frac{2x}{1-x^2}\right) + \tan^{-1}\left(\frac{3x-x^3}{1-3x^2}\right) = \frac{\pi}{4}$ is

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \int_0^x \frac{t^2 - at + 1}{(t^2+1)^2} \, dt$ Which of the following represents the set of all real values of $a$ for which $f$ is bijective?

Let $A = \begin{pmatrix} 4-i & -2+2i \\ 1 & 1 \end{pmatrix}$ and $I$ be the $2 \times 2$ identity matrix. A linear transformation $T_x: \mathbb{C}^2 \to \mathbb{C}^2$ is defined as $T_x(\mathbf{v}) = (A - xI)\mathbf{v}$, where $x \in \mathbb{R}$. If $S$ is the set of all real values of $x$ for which $T_x$ is NOT an injective mapping, then $S$ is equal to:

Let $f: \left[0, \frac{2}{\pi}\right] \rightarrow \left[0, \frac{2}{\pi} + \frac{4}{\pi^2}\right]$ be defined as: $f(x) = \begin{cases} x + x^2 \sin\left(\frac{1}{x}\right), & x \in \left(0, \frac{2}{\pi}\right] \\ 0, & x = 0 \end{cases}$ Consider the following statements: I. $f$ is a surjective function. II. $f$ is strictly increasing on its domain. III. $f$ is not differentiable at $x=0$ because $\lim_{x \to 0^+} f'(x)$ does not exist.

Which of the following represents the correct truth values of (I, II, III)?

Let $U_n = \{z \in \mathbb{C} : z^n = 1\}$. For each integer $n \ge 3$, define the mapping $f_n: U_n \to U_n$ by $f_n(z) = z^5 \exp\left( \frac{1}{2} (\text{Log}(z) + \text{Log}(\bar{z})) \right)$ where $\text{Log}(z)$ denotes the principal branch of the complex logarithm, with its imaginary part in $(-\pi, \pi]$. Let $S$ be the set of all integers $n \in \{10, 11, \dots, 20\}$ such that $f_n$ is a bijective mapping. Find the sum of the elements of $S$.

Let $f: \mathbb{R} \to S$ be a surjective, twice-differentiable function satisfying the differential equation $e^{2f(x)} (f'(x))^2 - 2x e^{f(x)} f'(x) + x^2 (1 - e^{2x}) = 0$ for all $x \in \mathbb{R}$, with $f(0) = 0$. Which of the following sets represents $S$?

Let $f: \mathbb{N} \to \mathbb{Z}$ be defined by $f(n) = \left\lfloor \sum_{k=1}^n \log_3 \left( 1 + \frac{107}{k^2 + 108k} \right) \right\rfloor$ where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Let $S$ be the range of $f$. Which of the following represents the cardinality of $S$?

Let $M_n(\mathbb{R})$ be the set of all $n \times n$ real matrices ($n \ge 2$). Define a mapping $f: M_n(\mathbb{R}) \to M_n(\mathbb{R})$ by $f(A) = A^T A$. Let $V = \{A \in M_n(\mathbb{R}) \mid A^2 = I_n\}$ be the set of all involutory matrices, and let $W = f(V)$ be the image of $V$ under $f$. Which of the following statements is ALWAYS true?

Let $A$ be a $3 \times 3$ skew-symmetric matrix with real entries. Let $f(x) = \det(A - xI)$. If the function $g: \mathbb{R} \to \mathbb{R}$ defined by $g(x) = f(x) + 245|x - \det(A)|$ is bijective, then the minimum possible value of $\text{Tr}(A^T A)$ is

Consider the function $f: [-1, \infty) \to \left[-\frac{1}{3}, \infty\right)$ defined by $f(x) = \frac{x^3}{3} + 2x^2 + 3x + \sum_{n=1}^{\infty} \frac{x^2}{(1+x^2)^n}$ Which of the following statements is true?

Let $S = \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}$. A function $f: S \to \mathbb{R}$ is defined such that $f(x, y, z)$ is the square of the length of the orthogonal projection of the vector $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$ onto the plane $x+y+z=0$. If $L$ is the total length of the curve(s) forming the pre-image $f^{-1}\left(\frac{1}{2}\right)$, find the value of $\frac{L^2}{\pi^2}$.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \left| \ln \left( \frac{x^2 - x + 1}{x^2 + x + 1} \right) \right|$. The set of all real values of $k$ for which the equation $f(x) = k$ has exactly $4$ distinct real solutions is:

Let $M_2(\mathbb{R})$ be the vector space of all $2 \times 2$ real matrices and $S = \{X \in M_2(\mathbb{R}) \mid X = X^T\}$ be the subspace of symmetric matrices. For a fixed matrix $A \in M_2(\mathbb{R})$, define linear transformations $T_1, T_2 : M_2(\mathbb{R}) \to M_2(\mathbb{R})$ by $T_1(X) = AX - XA$ $T_2(X) = AX - XA^T$ Let $k_1 = \min_{A \in M_2(\mathbb{R})} (\dim(\ker T_1))$ and $k_2 = \min_{A \in M_2(\mathbb{R})} (\dim(\ker (T_2|_S)))$, where $T_2|_S$ is the restriction of $T_2$ to $S$. What is the value of $k_1^2 + k_2^2$?

Let $A = \begin{pmatrix} 2 & 2 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -6 \\ 1 & 2 \end{pmatrix}$. For any $2 \times 2$ matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, define the transformation $T_M(x) = \frac{ax + b}{cx + d}$. Let $f(x) = T_A(x)$ and $g(x) = T_B(x)$. Let $S$ be the set of all real numbers $x$ satisfying the equation $f(g(h(x))) = g(f(h(x)))$, where $h(x) = \sqrt{|[x]|}$ and $[ \cdot ]$ denotes the greatest integer function. Find the value of $\int_{S} (3x^2 + 202x) \, dx$

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be defined as: $f(x) = \frac{x^2 - x + 1}{x^2 + x + 1}$ $g(x) = x^3 - 3x^2 + 3x + 2$ Let $A = [\alpha, \beta]$ be the range of $h(x) = g(f(x))$ and $B = [\gamma, \delta]$ be the range of $k(x) = f(g(x))$. Let $n_1$ be the number of real roots of the equation $h(x) = 4$, and $n_2$ be the number of real roots of the equation $k(x) = 1$. Find the value of $27\alpha + \beta + 3\gamma + \delta + n_1 + n_2$.

Let $f: \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \to \mathbb{R}$ be defined by $f(x) = \begin{cases} \frac{x^2}{\sin x} + x^2 \sin\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$ Let $g(x) = \frac{2}{\pi} - x$. Consider the composite functions $H_1(x) = f(g(x))$ and $H_2(x) = g(f(x))$. Which of the following statements is TRUE regarding the existence of local inverses for $H_1$ and $H_2$ in some neighborhood $(-\delta, \delta)$ of $x = 0$ (where $\delta > 0$ is sufficiently small)?

Let $f(x) = x + \frac{1}{x}$ and $g(x) = e^x$. Define the integrals $I_1 = \int_{1/2}^{2} f(g(x)) \, dx$ and $I_2 = \int_{1/2}^{2} g(f(x)) \, dx$. Which of the following inequalities is correct?

Let $\alpha, \beta$ (where $\alpha < \beta$) be the roots of the equation $x^2 - 7x + 12 = 0$. Let $f(x) = \alpha x - [\alpha x]$ and $g(x) = x + \frac{1}{\beta}$, where $[t]$ denotes the greatest integer less than or equal to $t$. Two sequences $\{x_n\}$ and $\{y_n\}$ are defined as follows: $x_1 = 0, \quad x_{n+1} = f(g(x_n)) \text{ for } n \ge 1$ $y_1 = 0, \quad y_{n+1} = g(f(y_n)) \text{ for } n \ge 1$ Find the value of $\sum_{n=1}^{120} (x_n + y_n)$.

Let $f: \mathbb{R} \to (0, \infty)$ be a differentiable function satisfying the differential equation $\frac{dy}{dx} + y = y^3 \left( 1 + \frac{3-2x}{4}e^{2x} \right)$ If $f(x)$ is an injective function on $\mathbb{R}$, then the maximum possible value of $f(0)$ is:

Let $S$ be the set of all sequences $(x_1, x_2, x_3)$ of positive integers such that $x_1 + x_2 + x_3 = 92$. A function $f: S \to \mathbb{Z}$ is defined by $f(x_1, x_2, x_3) = x_1 + 2x_2 + 3x_3$. If $A \subseteq S$ is the largest possible subset such that the restriction $f|_A$ is injective, find the cardinality of $A$.

Let $L_1$ and $L_2$ be two lines in $\mathbb{R}^3$ given by: $L_1: \vec{r} = \lambda(\hat{i} + 2\hat{j})$ $L_2: \vec{r} = (\hat{i} - 2\hat{j} + \hat{k}) + \mu(2\hat{j} + \hat{k})$ Let $d$ be the shortest distance between $L_1$ and $L_2$. Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = x^3 - 3dx^2 + 3(3d-2)x$. Consider the vectors $\vec{u} = x\hat{i} + \hat{j} + \hat{k}$, $\vec{v} = \hat{i} + x\hat{j} + \hat{k}$, and $\vec{w} = \hat{i} + \hat{j} + x\hat{k}$. A function $g: \mathbb{R} \to \mathbb{R}$ is defined as: $g(x) = \begin{cases} \frac{[\vec{u} \ \vec{v} \ \vec{w}]}{|\vec{u} \times \vec{v}|^2} & \text{if } \vec{u} \text{ and } \vec{v} \text{ are non-collinear} \\ k & \text{if } \vec{u} \text{ and } \vec{v} \text{ are collinear} \end{cases}$ If $g(x)$ is continuous for all $x \in \mathbb{R}$, which of the following is true?

Let $M = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ and $S = \{ aM + bM^2 \mid a, b \in \mathbb{R} \}$. A binary operation $*$ on $S$ is defined by $A * B = A + B + c AB$, where $c$ is a non-zero real constant. For any $A \in S$, let $A^{\circ n} = A * A * \dots * A$ ($n$ times). If $X \in S$ is the inverse of $M^{\circ 5}$ under the operation $*$, then $X$ is equal to:

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. A binary operation $*$ on the power set $\mathcal{P}(S)$ is defined as $A * B = (A \setminus B) \cup (B \setminus A)$. Let $\mathcal{I}$ be the family of all proper subsets of $S$ that do not contain any two consecutive integers. If $\mathcal{C}$ is the closure of $\mathcal{I}$ under the operation $*$, find the number of elements in the set $\mathcal{V} = \{ A \in \mathcal{C} \setminus \{S\} \mid A \notin \mathcal{I} \}$.

Let a binary operation $*$ be defined on a subset $S \subseteq \mathbb{R}$ as $a * b = \left[ \text{sgn}(a) \cos^2\left(\frac{\pi b}{2}\right) \right]$ where $[\cdot]$ denotes the greatest integer function and $\text{sgn}(\cdot)$ denotes the signum function. For which of the following sets $S$ is $*$ a closed binary operation?

Let $B = \{ \vec{v} \in \mathbb{R}^3 : |\vec{v}| \le 1 \}$ be the closed unit ball in $\mathbb{R}^3$. A binary operation $\star$ is defined on $B$ by $\vec{u} \star \vec{v} = \lambda (\vec{u} \cdot \vec{v}) (\vec{u} \times \vec{v})$ where $\lambda > 0$ is a real constant. What is the maximum value of $\lambda$ such that $B$ is closed under the operation $\star$?

Let $S$ be the set of all $2 \times 2$ matrices of the form $M(x, y) = \begin{pmatrix} x & y \\ 0 & \frac{1}{x} \end{pmatrix}$, where $x \in \mathbb{R} \setminus \{0\}$ and $y \in \mathbb{R}$. A binary operation $\star$ is defined on $S$ by $A \star B = A B A^{-1}$ for all $A, B \in S$. Let $L$ be the set of all left identities and $R$ be the set of all right identities for the operation $\star$ in $S$. Which of the following statements is correct?

Let $S = \mathbb{R} \setminus \{ (2k+1)\pi : k \in \mathbb{Z} \}$. A binary operation $*$ on $S$ is defined as $x * y = 2 \tan^{-1} \left( \tan \frac{x}{2} + \tan \frac{y}{2} \right)$ Consider the following three statements: $S_1$: The operation $*$ admits an identity element in $S$. $S_2$: The operation $*$ is associative on $S$. $S_3$: The function $f: S \to (-\pi, \pi)$ defined by $f(x) = x * 0$ is surjective but not injective. Which of the following is correct?

Let $S = \mathbb{R} \setminus \{\alpha\}$. A binary operation $*$ is defined on $S$ by $x * y = x y - \lambda x - \lambda y + \lambda^2 + \lambda$ where $\lambda = \sum_{n=2}^{\infty} \frac{2n+6}{n^3-n}$. If the operation $*$ admits a unique identity element in $S$, what is the value of $\alpha$?